Many elements of building structures (columns, racks, supports) are under the influence of compressive forces attached not in the center of severity. In fig. 12.9 shows a column on which the beam of overlapping is based on. As can be seen, the force acts in relation to the axis of the column with the eccentricity e, And thus, in an arbitrary section a-A. Columns along with longitudinal force N. = -R There is a bending moment, the magnitude of which is equal Re. Energy stretching (compression) The rod represents a type of deformation, in which the equal external forces act along the straight, parallel string axis. In the future, we will consider mainly the tasks of the extracentar compression. With an off-centrular stretching in all given calculated formulas, change the sign before the force R On the opposite.

Let the terminal of the arbitrary cross section (Fig. 12.10) are loaded on the end of the extracentrately applied compressive force R, directed parallel to the axis Oh. Let's take positive

directions of the main axes of the cross section OU and Oz. So that the point of application of force R It was located in the first quarter of the axes of coordinates. Denote the coordinates of the Power Appointment point R through r. and z p -

Internal efforts in an arbitrary cross section of the rod equal

The signs of minus in bending moments are due to the fact that in the first quarter of the axes of coordinates, these moments cause compression. The values \u200b\u200bof internal efforts in this example are not changed along the length of the rod, and thus, the distribution of stresses in sections, sufficiently deleted from the location of the application application will be the same.

Substituting (12.11) in (12.1), we obtain a formula for normal stresses during an off-centrular compression:

This formula can be converted to mind

where i, I- Main radii inertia section. Wherein

Putting in (12.12) O \u003d 0, we obtain the equation zero line:

Here y 0 and z 0 - The coordinates of the points of the zero line (Fig. 12.11). Equation (12.14) is a direct equation that does not pass through the severity center. To make a zero line, we will find the points of its intersection with the axes of coordinates. Believing in (12.14) sequentially y 0 \u003d 0 and z 0 \u003d 0, respectively, find

where a Z. and and y - Cuts cut off zero line on the coordinate axes (Fig. 12.11).

We will establish the characteristics of the position of the zero line with a non-centered compression.

• 1. From formulas (12.15) it follows that a W. and a Z. have signs opposite to signs respectively r. and z p - Thus, the zero line passes through those quarters of the coordinate axes that do not contain the point of the application of force (Fig. 12.12).
• 2. With the approach of the point of the application of force R For a straight line to the center of severity section of the coordinate of this point r. and z P. decrease. From (12.15) it follows that the absolute values \u200b\u200bof the lengths of the segments a W. and a Z. Increase, that is, the zero line is removed from the center of gravity, remaining parallel to itself (Fig. 12.13). In the limit of Z p \u003d y p \u003d 0 (Power is applied in the center of gravity) The zero line is removed in infinity. In this case, in the stress cross section will be permanent and equal to \u003d -P / f.
• 3. If the point of power application R It is located on one of the main axes, the zero line is parallel to another axis. Indeed, putting in (12.15), for example, r. \u003d 0, we get that a W. \u003d that is, the zero line does not cross the axis OU (Fig. 12.14).
• 4. If the point of the application application moves in a straight line, not passing through the center of gravity, the zero line turns around a certain point. We prove this property. Points of application forces R H. and P 2, located on the coordinate axes correspond to the zero lines 1 - 1 and 2-2, parallel to the axes (Fig. 12.15), which intersect at the point D. Since this point belongs to two zero lines, the voltages at this point from the simultaneously attached forces R H. and P 2. will be zero. Since any power P 3,the point of application of which is located on a straight P (P 2, can

decompose on two parallel components attached at PJ and P 2, then hence it follows that voltages at the point D.from the action of power P 3. Also equal to zero. Thus, the zero line 3-3, corresponding to force P 3, passes through the point D.

In other words, a variety of points R,located on direct P (P 2, corresponds to a beam of direct passing, through the point D. Fair and reverse statement: when the zero line is rotated around a certain point, the point of the force application moves in a straight line, not passing through the center of gravity.

If the zero line crosses the section, then it divides it into compression zones and stretching. As well as in oblique bending, from the hypothesis of flat sections it follows that the voltages reach the greatest values \u200b\u200bat the points most remote from the zero line. The nature of the stress plots in this case is shown in Fig. 12.16, but.

If the zero line is located outside the section, then at all points of the voltage section there will be one sign (Fig. 12.16, b).

Example 12.3. We construct the stage of normal stresses in an arbitrary cross section of an echocently compressed rectangular cross-section column b. H. h. (Fig. 12.17). The squares of the inertia radii of the cross section according to (12.22) are equal

Segments that cut by zero line on the axes of coordinates are determined by formulas (12.15):

Substituting sequentially in (12.12) the coordinates of the points from the null line from and IN (Fig. 12.18)

find

Epur about is shown in Fig. 12.18. The largest compressive voltages in absolute value are four times higher than voltages, which would be in the case of a central application of force. In addition, significant tensile stresses appeared in the cross section. Note that from (12.12) it follows that in the center of gravity (y \u003d z \u003d 0) voltages are equal to \u003d -P / f.

Example 12.4. The strip with a cut is loaded with stretching power R (Fig. 12.19, but). Compare stresses in cross section Lv sufficiently removed from the end and the place of the cut, with stresses in the section CD In the cutting place.

In cross section AU (Fig. 12.19, b) force R Causes central tension and voltage equal to \u003d P / F \u003d P / BH.

In cross section CD (Fig. 12.19, in) Power line line R It does not pass through the severity center, and therefore an extracentrate stretch arises. Changing the Sign in Formula (12.12) to the opposite and accepting r. \u003d 0, we get for this section

Taking

Zero line in cross section CD parallel to axis OU and crosses the axis Oz. on distance a \u003d.-i 2 y / z p- b /12. In the sections of the section most remote from the zero line C (Z - -B /4) I. D (Z - b /4) voltages according to (12.16) are equal

Maps of normal stresses for sections Lv and CD Showing in Fig. 12.19, b, c.

So, despite the fact that the cross section CD has an area of \u200b\u200btwo times smaller than the cross section AB Due to the extracentrate application forces, the stretching voltages in a weakened cross section increases not in two, but eight times. In addition, in this section there are significant compressive stresses.

It should be noted that the above calculation does not take into account additional local stresses arising near the point C because of the presence of a shading. These voltages depend on the radius of the pump (with a decrease in the radius they increase) and can significantly exceed the value in the value a C. = 8P / BH.In this case, the nature of the stress plots near the point C will differ significantly from linear. Definition of local stresses (stress concentration) is considered in chapter 18.

Many building materials (concrete, brickwork, etc.) badly resist stretching. Their tensile strength is many times less than compression. Therefore, in elements of structures from such materials, the appearance of tensile stresses is undesirable. So that this condition is performed, it is necessary that the zero line is out of section. Otherwise, the zero line will cross the section and stretching stresses will appear. If the zero line is tangent to the section contour, then the corresponding position of the point of the application of force is the limit. In accordance with the properties of the 2 zero line, if the point of the application of force will approach the severity center, the zero line will be removed from it. The geometric location of the limit points corresponding to various concerns to the contour of the section is the boundary core sections. The core of the section is called a convex area around the center of gravity, which has the following property: if the point of the application of the force is inside or on the border of this area, then in all points of the voltage cross section have one sign. The core of the section is a convex figure, since zero lines should touch the cross section of the cross section and do not cross it.

Through the point BUT (Fig. 12.20) You can conduct countless tangential (zero lines); At the same time only tangent AC It is a tangent to the envelope, and it must correspond to a certain point of the cross section of the cross section. At the same time, for example, it is impossible to conduct a tangent AU Contour of the section, because it crosses the cross section.

We construct the core of the cross section for the rectangle (Fig. 12.21). For tangent 1 - 1 a 7 - b /2; but \u003d. From (12.15) we find for point 1 corresponding to this tangential z p \u003d -i 2 y / a 7 \u003d -b / 6; at r - 0. For tangent 2-2 and y - to /2; a 7 \u003d °°, and the coordinates of the point 2 will be equal w.r- -H / 6; z p - 0. According to the property of the 4 zero line of the point of the application of the force corresponding to the different tangent of the right-lower angular point of section, are located on a direct 1-2. The position of the points 3 and 4 is determined from the conditions of symmetry. Thus, the core of the cross section for the rectangle is a diagonal rhombus B/ 3 I. OF.

To build a sequence core for a circle, it is enough to carry out one tangent (Fig. 12.22). Wherein a \u003d r; but \u003d ° o.

"Y ^ ^

Considering that for the circle i - j y / f - r /4, from (12.15) we get

Thus, the core of the cross section for the circle is a circle with a radius R / 4.

In fig. 12.23, a, 6. Showing cross sections for a hectare and a schuelevra are shown. The presence of four angular points of the sections in each of these examples is due to the fact that the envelope of the contour and the inlet of the alter and the schveller is a rectangle.

The extracentral stretching (compression) is caused by the force parallel to the axis of the bar, but not coinciding with it (Fig. 9.4).

The projection of the point of the application of force on the cross section is called a pole or power point, and a straight, passing through the pole and the center of the section - the power line.

Especially stretching (compression) can be reduced to axial stretching (compression) and oblique bending, if you transfer the force of the force of the severity. So, the force p, marked in Fig. 9.4 One dash r will cause axial stretching of the bar, and a pair of forces marked with two dashes is oblique bending.

Based on the principle of independence of the voltage forces at the cross-sectional points during incantented tension (compression) are determined by the formula

In this formula, the axial force of the bending moments as well as the coordinates of the point of section, in which the voltage is determined, it is necessary to substitute with their signs. For bending moments, we will take the same rule of signs, as in oblique bending, and we will consider the axial strength positive when it causes a stretching.

If the coordinates of the pole designate through, then the moment of formula (9.5) takes the view

It can be seen from this equation that the ends of the voltage vectors at the sections are located on the plane. The line intersection line with a cross-sectional plane is a neutral line, the equation of which we find, equating the right-hand side of the equality (9.6) zero. After cutting on the p we get

Thus, the neutral line with non-centrular tension (compression) does not pass through the center of severity and is not perpendicular to the plane of the bending moment. The neutral line cuts off on the axes of the segments coordinates

Imagine the moments of inertia as the work of the cross-section area on the square of the corresponding radius of inertia

Then the expressions (9.8) can be written as follows:

From formulas (9.8) it can be seen that the pole and the neutral line are always located along different directions from the center of secting, the position of the neutral line is determined by the coordinates of the pole.

When the pole approaches the power line to the severity center, the neutral line will be removed from the center, while remaining parallel to its original direction. In the limit with a neutral line will be removed in infinity. In this case, there will be a central stretching (compression) of the bar.

On the power line you can always find this position of the pole, at which the neutral line will touch the contour of the section, without crossing it anywhere. If you have all possible neutral lines so that they concern the section circuit, without crossing it, and find them the corresponding poles, it turns out that the poles will be located on a closed line quite defined for each cross section. The area bounded by this line is called the core of the section. In the circular section, for example, the kernel is a circle of diameter of 4 times smaller diameter of the section, and in rectangular and altar sections, the kernel has a parallelogram form (Fig. 9.5).

From the very construction of the cross section, it follows that as long as the pole is inside the nucleus, the neutral line will not cross the circuit of the section and the voltage throughout the cross section will be one sign. If, the pole is located outside the nucleus, the neutral line will cross the section circuit, and then in the section there will be voltages of different signs. This circumstance must be taken into account when calculating the subsistence compression of racks from fragile materials. Since fragile materials are poorly perceived by tensile loads, it is preferably the external forces to apply to the rack so that only compression voltages acted throughout the cross section. For this purpose, the point of the application of the equal external forces compressing the rack must be inside the cross section core.

The calculation of the strength during non-centrular stretching and compression is made in the same way as in the oblique bending - by voltage at a dangerous point of the cross section. The cross section is dangerous, the most remote from its neutral line. However, in cases where the compression voltage is acting at this point, and the rack material is fragile, the dangerous point in which the largest tensile voltage is acting.

The stress plot is built on the axis perpendicular to the neutral cross section line, and is limited to the straight line (see Fig. 9.4).

The condition of strength will be recorded so.

where ; ;

y f, z f- Coordinates of the point of the application of force F..

To determine the dangerous points of section, it is necessary to find the position of the neutral line (N.L.) as a geometric location of points in which the voltages are zero.

.

Equation N.L. It can be recorded as a straight equation in segments:

where and - Cut, cut off N.L. on the axes of coordinates

- The main radii of the inertia of the section.

The neutral line divides the cross section on zones with tensile and compressive stresses. Eppure of normal stresses is presented in Fig. 8.4.

If the cross section is symmetrically relative to the main axes, the condition of strength is written for plastic materials, which [ s C.] = [s P.] = [s.], as

. (8.5)

For fragile materials, which [ s C.]¹[ s P.], the condition of strength should be recorded separately for a dangerous section of the cross section in the stretched zone:

and for a dangerous point of section in a compressed zone:

,

where z 1., y 1. and z 2., y 2. - The coordinates of the secting points are most distant from the neutral line in a stretched 1 and compressed 2-sided cross sections (Fig. 8.4).

Non-line properties

1. The zero line divides all the cross-section into two zones - stretching and compression.

2. The zero line is direct, since the coordinates x and y first.

3. The zero line does not pass through the origin of the coordinates (Fig. 8.4).

4. If the point of the application of force lies on the main central inertia of the section, then the corresponding zero line is perpendicular to this axis and passes on the other side of the coordinates (Fig. 8.5).

5. If the point of the application of the force moves along the beam, outgoing from the beginning of the coordinates, the corresponding zero line moves behind it (Fig. 8.6):

Fig. 8.5 Fig. 8.6.

a) when the point of the point of application of force on the beam, originating from the beginning of the coordinates from zero to infinity (Y F ®∞, Z F ®∞), but ®0; but z ®0. The limit state of this case: the zero line will be held through the origin of the coordinates (bending);

b) when the point of application of force (T. K) is moving along the beam, outgoing from the beginning of the coordinates from infinity to zero (Y F ® 0 and Z F ® 0), but ®∞; but z ®∞. The limit state of this case: the zero line is removed in infinity, and the body will experience a simple stretching (compression).

6. If the point of the force application (T. K) moves in a straight line, crossing the coordinate axes, then in this case the zero line will rotate around some center located in the opposite point to the quadrant.

8.2.3. Core section

Some materials (concrete, brickwork) can perceive very minor tensile stresses, while others (for example, soil) cannot resist stretching at all. Such materials are used to manufacture structural elements in which tensile voltages do not occur, and are not used for the manufacture of elements of instructions experiencing bending, tapping, central and high-centered stretching.

From these materials, it is possible to produce only centrally compressed elements in which the tensile stresses do not arise, as well as high-centeredly compressed elements, if tensile voltages are not formed in them. This occurs when the point of application of the compressive force is located inside or at the boundary of some central cross-sectional area, called the core of the section.

Core section Bruus is called his some central region, which has the property that the force attached at any of its point causes in all points of the cross section of the voltage bar of one sign, i.e. The zero line does not pass through the cross section of the bar.

If the point of the compressive force application is located outside the core of the section, compressing and tensile stresses occur in cross section. In this case, the zero line crosses the cross section of the bar.

If the force is applied on the border of the secting core, the zero line concerns the cross section contour (at the point or line); In touch site, normal stresses are zero.

When calculating non-centerfully compressed rods made of material, poorly perceiving tensile voltages, it is important to know the shape and size of the cross section of the cross section. This allows, without calculating the stresses, to establish whether the tensile stresses arise in the cross section (Fig. 8.7).

From the definition it follows that the core of the section is some area that is inside the cross section itself.

For fragile materials, the compressive load should be applied in the cross section core to exclude in the cross section of the stretching zone (Fig. 8.7).

To construct the core of the section, it is necessary to consistently combine the zero line with the cross-sectional circuit in such a way that the zero line does not restore the section, and at the same time count the corresponding point

compressive force applications to with coordinating

Fig. 8.7 Dinatama y F and z F. By formulas:

; .

The resulting points of the application of force with coordinates y f, z f It is necessary to connect the sections of the straight lines. The area bounded by the resulting broken line will be the core of the section.

Sequence of cross section core

1. Determine the position of the center of gravity of the cross section and the main central axes of inertia y and Z., as well as the values \u200b\u200bof the squares of inertia radii i. y, I z.

2. Show all possible positions N.L., concerning the cross section contour.

3. For each position N.L. Determine the segments a Y. and a Z.that cut off from the main central axes of inertia y and z.

4. For each position N.L. Install the coordinates of the pressure center y F, I. z F. .

5. The obtained pressure centers are connected to the sections of direct, inside which the core of the section will be located.

Crash with bending

The type of loading at which the timber is exposed simultaneously by the action of twisting and bending moments, is called a bend with a mold.

When calculating, we use the principle of independence of the forces. We define the stresses separately under bending and drying (Fig. 8.8) .

When bending in cross section, normal stresses arise, reaching the maximum value in extreme fibers.

.

When tangent in cross section, tangent stresses arise, reaching the greatest value at the sections of the section at the surface of the shaft

.

Normal and tangent stresses simultaneously reach the greatest value at points. FROMand IN Section of shaft (Fig. 8.9). Consider the stressful state at the point. FROM(Fig. 8.10). It can be seen that elementary parallelepiped, highlighted around the point FROMis at a flat stress state.

Therefore, to test the strength, one of the hypotheses of strength is applicable.

The condition of strength on the third hypothesis of strength (hypothesis of the largest tangent stresses)

.

Considering that , We obtain the condition of the strength of the shaft

. (8.6)

If the bending of the shaft occurs in two planes, then the condition of strength will be

.

Using the fourth (energy) strength hypothesis

,

after substitution s. and t.receive

. (8.7)

Questions for self-test

1. What is the bending called oblique?

2. What kind of bend is the combination of bending?

3. What formulas are the normal stresses in the cross sections of the beam beams?

4. How is the position of the neutral axis with oblique bending?

5. How do dangerous dots define in cross section during oblique bending?

6. How are the movement of the axis points of the axis of the beam with oblique bending?

7. What kind of complex resistance is called an outcidentren stretching (or compression)?

8. What formulas are the normal stresses in the cross sections of the rod during non-centcenary stretching and compression? What kind of is the epira of these stresses?

9. How determines the position of the neutral axis during non-centrular stretching and compression? Record the appropriate formulas.

10. What stresses occur in the cross section of the bar when bending with a clearance?

11. How are there dangerous sections of a round-section timber when bending with a twist?

12. What are the points of the circular cross section are dangerous when bending with a mold?

13. What stress state occurs at these points?